A new efficient method that combines the Puiseux series asymptotic technique with an augmented compact finite volume method is proposed to develop a numerical approximate solution for the Thomas–Fermi equation on semi-infinity domain. By using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularities, the natural and precise boundary conditions are obtained. The expansions contain undetermined parameters which associate with the singularity as the augmented variables. A regular boundary value problem is derived, for which an augmented compact finite volume method is used. The computational results show that the method not only obtains the high precise numerical solution, but also obtains the high precise initial slope. In particular, we find that the initial slope is exactly equal to the augmented variable related to the singularities in the Puiseux series. The initial slope not only has an important physical significance, but also its calculation accuracy has become an important criteria to measure the quality of the algorithm.

提出了一种新的有效方法，将 Puiseux 级数渐近技术与增广紧致有限体积方法相结合，以开发半无限域上 Thomas-Fermi 方程的数值近似解。利用无穷远解的渐近级数和原点Puiseux级数展开刻画奇点，得到自然而精确的边界条件。扩展包含未确定的参数，这些参数与作为增广变量的奇点相关联。导出了一个正则边值问题，为此使用了增广紧致有限体积方法。计算结果表明，该方法不仅得到了高精度的数值解，而且得到了高精度的初始斜率。特别是，我们发现初始斜率恰好等于与 Puiseux 级数中奇点相关的增广变量。初始斜率不仅具有重要的物理意义，其计算精度也成为衡量算法优劣的重要标准。